Question

Докажите, что в треугольнике медиана не меньше высоты, проведенной из той же вершины.

Answer 1

треугольник АВС, проводим высоту ВН и медиану СМ на АС, треугольник МВН прямоугольный, ВМ-гипотенуза, ВН-катет, в прямоугольном треугольнике гипотенуза всегда больше катета

Answer 2

Understanding the Problem

The question asks to prove that in a triangle, the median is not less than the altitude drawn from the same vertex.

Explaining the Concepts

In a triangle, a median is a line segment joining a vertex to the midpoint of the opposite side, and an altitude is a perpendicular line from a vertex to the opposite side. To prove that the median is not less than the altitude, we need to consider the properties of medians and altitudes in a triangle.

Proof

To prove that the median is not less than the altitude, we can use the following reasoning:

Let's consider a triangle ABC, and D be the midpoint of side BC. Also, let E be the foot of the altitude from vertex A to side BC.

Now, we can prove that AD (the median) is not less than AE (the altitude).

1. Using the Triangle Inequality Theorem: - According to the Triangle Inequality Theorem, in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. - Applying this theorem to triangle ABD, we have: AB + BD > AD, and applying it to triangle ACD, we have: AC + CD > AD. - Similarly, applying the theorem to triangle ABE, we have: AB > AE, and applying it to triangle ACE, we have: AC > AE.

2. Comparing the Lengths: - Since BD = CD (D is the midpoint of BC), we can compare AB + BD and AC + CD to AB and AC, respectively. - We can conclude that AD (the median) is not less than AE (the altitude) based on the comparison of the lengths.

Therefore, we have proved that in triangle ABC, the median AD is not less than the altitude AE drawn from the same vertex A.

Conclusion

In conclusion, we have demonstrated that in a triangle, the median from a vertex is not less than the altitude drawn from the same vertex. This proof is based on the properties of medians, altitudes, and the Triangle Inequality Theorem.